Understanding of Combination

Understanding of combinations (aka. Selections)

combination is to select things from a given set of things without considering sequences. Selecting \(r\) items out of a group of \(n\) items is denoted as \(C_n^r\).

The combination formula is \(C_n^r = \frac{n!}{r!(n-r!)}\) where \(0\leq r\leq n\).

To select 3 things out of 5 things with order, there are \(5\times4\times3=60\) permutations, namely \(\frac{5!}{(5-3!)}\).

Among these 3 selected things, there are \(3\times2\times1=6\) permutations, namely \(3!\).

Thus we can solve combination problems with permutation problems! The number of combination that selecting 3 things out of 5 without order is \(\frac{5!}{3!(5-3!)}\) denoted as \(\frac{n!}{r!(n-r!)}\).